Atomic Forces and Phase Transitions

Jacob Kelter
Chemistry, Physics
1 hour 30 minutes
High School Grade 12
v1

Overview

Students will explore phase transitions and packing structure using a model of atoms based on the Lennard-Jones potential.

Standards

Next Generation Science Standards
  • Physical Science
  • NGSS Crosscutting Concept
    • Patterns
    • Causation
    • Energy
    • Systems
    • Stability and Change
    • Structure and Function
  • NGSS Practice
    • Constructing Explanations, Designing Solutions
    • Asking Questions, Defining Problems
    • Using Models
    • Arguing from Evidence
    • Conducting Investigations
Computational Thinking in STEM
  • Modeling and Simulation Practices
    • Assessing Computational Models
    • Using Computational Models to Understand a Concept
  • Systems Thinking Practices
    • Understanding the Relationships within a System

Activities

  • 1. Lennard-Jones Model
  • 2. Going Further Option 1 - Single Type of Atom (More coding less chemistry)
  • 3. Going Further Option 2 -Two Types of Atoms (More chemistry less coding)
  • 4. Discussion

Student Directions and Resources


In this lesson you will explore how atoms arrange themselves in solids and phase transitions using an agent based model. In an agent based model, small entities called "agents" follow simple computational rules, and then complex patterns emerge from their interactions. In this case, the agents are going to be atoms, and the emergent behaviors will be how they arrange themselves in solids and phase transitions.

1. Lennard-Jones Model


Start by playing around with the model and then use it to answer the questions below.

 


Question 1.1

Estimate temperature ranges when the system is clearly a solid and clearly a gas. It is hard to tell in a 2-dimensional model when the atoms are in a liquid, but try to estimate the temperature range for liquid as well.

Use your ranges to estimate the melting point and the boiling point. 



Question 1.2

As you change the temperature, what happens to the average energy per atom? Why do you think that is?



Question 1.3

Use some basic NetLogo commands to figure out what the maximum and minimum energies are for atoms in the model. You'll type the commands in the Command Center underneath the model.

Here are some useful commands:

  • [calc-v] of turtles: this returns a list of the energies for all the atoms
  • min(...) and max(...): you have to put a list inside the parentheses

(hint:  you shouldn't have to copy and paste anything. You just have to write the right command.)

Does the temperature affect the the maximum and minimum energies?



Question 1.4

This is a graph of equation used to model potential energy between two atoms used in the agent based model. It is known as the Lennard-Jones potential after Sir John Edward Lennard-Jones who came up with it. The equation is:

Solve for rminin terms of ε and σ. Ask for a hint if you aren't sure how to do that.



Question 1.5

What is the minimum potential energy between two atoms in terms of variables in the Lennard-Jones potential?



Question 1.6

When the temperature is low enough for the atoms to solidify, how many neighbors can an atom have? Why is this?



Question 1.7

Calculate potential energy of an atom in the configuration from the previous question, assuming all of it's neighbors are rmin distance away.

When you used Netlogo commands to find the minimum energy of atoms, did any of them have this energy? Why? (to find out what the model uses for ε, go into the Netlogo Code tab and look for where it gets set. ε is called EPS in the code.)



Question 1.8

Add a slider to change the value ε.

To do that, click the lock button in the upper left corner that looks like this: , and it will change to authoring mode.

You can click and drag things to move them around. Right click to add a slider. Call it EPS, and set the values of the slider to what seems reasonable based on what EPS is currently set to. When you try to add the slider, you should get an error that says "There is already a global variable called EPS." So, go into the NetLogo code tab and find where the EPS variable is currently set. Either delete it, or turn it into a comment that the computer ignores by putting a semicolon in front of it. You also will need to delete EPS from the list of globals at the top of the code. 

No explore how changing your EPS slider changes the behavior of the model and describe what you find.

2. Going Further Option 1 - Single Type of Atom (More coding less chemistry)


If you have time, try some of the extensions and explorations below. You don't have to answer them in order, or answer all of them.


Question 2.1

Try starting the model in HCP (this stands for hexagonally close packed) and in random configurations. When the configuration starts random, why can't you ever get all the atoms in one cluster the way they are when they all start in HCP?



Question 2.2

In this question, you will have atoms change their color based on their energy.

First play around with changing the color of turtles. Try typing ask turtles [set color white] in the Command Center. Try whatever colors you want. You can also ask a single turtle to change its color by typing for example ask turtle 0 [set color red]. All the turtles have a number.

You can set the color to a different shade with the scale-color command. The whole command for a shade of a color is scale-color color number range1 range2. This picks a shade of color based on where number falls between range1 and range2. The closer number is to one end of the range, the lighter the output color will be and the closer it is to the other end of the range, the darker the output will be. Try using this to change the atoms colors to different shades.

When you are comfortable with that, go into the code tab and find where the turtles calculate their energy. Use the calculated energy to set their color. So each turtle should have a different color or shade of color based on its energy. This could be tricky, so feel free to ask for hints if you can't figure it out.

 

Do you notice any patterns in the energy of atoms?



Question 2.3

Give different atoms different sizes instead of them all having the same size. 

Explain the approach you used to do this and why you did it that way.

3. Going Further Option 2 -Two Types of Atoms (More chemistry less coding)


Now that you've seen the basic model with one type of atom, it's time to explore something a little more advanced. The model below basically behaves the same way, but there are two types of atoms. You don't have to answer the questions in order, or even answer them at all if you don't want to.

 


Question 3.1

There are now three different EPS parameters. EPS-AA is the epsilon value for the potential function between two A type atoms. EPS-BB is the epsilon value for the potential function between two B type atoms. EPS-AB is the epsilon value for the potential function between an A and B type atoms.

Play around with different values and see what types of behavior you get. Write down what you find and try explain why you see what you do. 

 



Question 3.2

Solutions are homogeneous mixtures of two or more pure substances. In a solution, the solute is dispersed uniformly throughout the solvent.

What values of EPS-AA, EPS-BB and EPS-AB result in a solution forming? If a solution doesn't form, what is forming? 

You can raise and lower the temperature to get the atoms to move around more and then settle back down. This can help reach an equilibrium faster. 



Question 3.3

Is the are the melting and boiling temperatures different depending on the values of EPS-AA, EPS-BB and EPS-AB? Does this relate to whether they form solutions at those values?



Question 3.4

In the real world, what types of inter-molecular forces could be responsible for EPS-AA, EPS-BB and EPS-AB being different relative strengths?



Question 3.5

If EPS-AA, EPS-BB and EPS-AB are all equal, then the atoms should mix randomly. Is uniform mixing the same as random mixing?



Question 3.6

Note: if you haven't done the other extra credit section yet, do that before doing the next two questions.

Try having the atoms scale their color based on their energy. Does this help you explain to yourself or others why you see the different behaviors you do with different values of EPS-AA, EPS-BB and EPS-AB?



Question 3.7

Add two sliders for the diameters of A and B atoms and change the code accordingly. Play around with making the atoms different sizes in addition to changing the values of EPS. What kind of behaviors do you get? Does this change the packing structure when the atoms are in a solid?

4. Discussion


Now we'll have a class discussion.


Question 4.1

Discussion Question 1



Question 4.2

Discussion Question 2



Question 4.3

Discussion Question 3